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January  2020, 40(1): 579-596. doi: 10.3934/dcds.2020023

Existence and nonexistence of subsolutions for augmented Hessian equations

School of Mathematics and Information Science, Weifang University, Weifang 261061, China

Received  April 2019 Published  October 2019

Fund Project: The author is supported by Shandong Provincial Natural Science Foundation, China (ZR2018LA006)

In this paper, we consider the augmented Hessian equations $ S_k^{\frac{1}{k}}[D^2u+\sigma(x)I] = f(u) $ in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $. We first give the necessary and sufficient condition of the existence of classical subsolutions to the equations in $ \mathbb{R}^{n} $ for $ \sigma(x) = \alpha $, which is an extended Keller-Osserman condition. Then we obtain the nonexistence of positive viscosity subsolutions of the equations in $ \mathbb{R}^{n} $ or $ \mathbb{R}^{n}_+ $ for $ f(u) = u^p $ with $ p>1 $.

Citation: Limei Dai. Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 579-596. doi: 10.3934/dcds.2020023
References:
[1]

J. G. BaoX. H. Ji and H. G. Li, Existence and nonexistence theorem for entire subsolutions of $k$-Yamabe type equations, J. Differential Equations, 253 (2012), 2140-2160.  doi: 10.1016/j.jde.2012.06.018.  Google Scholar

[2]

L. A. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, Ⅰ. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar

[3]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅲ. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

[4]

I. Capuzzo DolcettaF. Leoni and A. Vitolo, Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 147-161.   Google Scholar

[5]

I. Capuzzo DolcettaF. Leoni and A. Vitolo, On the inequality $F(x, D^2u)\geq f(u)+g(u)|Du|^q$, Math. Ann., 365 (2016), 423-448.  doi: 10.1007/s00208-015-1280-2.  Google Scholar

[6]

H. Car and R. Pröpper, Removable singularities of $m$-Hessian equations, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 6, 18 pp. doi: 10.1007/s00030-016-0429-3.  Google Scholar

[7]

K. S. Chou and X. J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016.  Google Scholar

[8]

D. P. Covei, The Keller-Osserman problem for the $k$-Hessian operator, arXiv: 1508.04653. Google Scholar

[9]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 219-245.  doi: 10.1016/S0294-1449(00)00109-8.  Google Scholar

[11]

S. DumontL. DupaigneO. Goubet and V. Radulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.  doi: 10.1515/ans-2007-0205.  Google Scholar

[12]

P. L. Felmer and A. Quaas, On critical exponents for the Pucci's extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 843-865.  doi: 10.1016/S0294-1449(03)00011-8.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

B. Guan, Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.  Google Scholar

[15]

Y. HuangF. D. Jiang and J. K. Liu, Boundary $C^ {2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.  doi: 10.1016/j.aim.2014.12.043.  Google Scholar

[16]

X. H. Ji and J. G. Bao, Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc., 138 (2010), 175-188.  doi: 10.1090/S0002-9939-09-10032-1.  Google Scholar

[17]

F. D. Jiang and N. S. Trudinger, On the Dirichlet problem for general augmented Hessian equations, arXiv: 1903.12410. Google Scholar

[18]

F. D. JiangN. S. Trudinger and X. P. Yang, On the Dirichlet problem for Monge-Ampère type equations, Calc. Var. Partial Differential Equations, 49 (2014), 1223-1236.  doi: 10.1007/s00526-013-0619-3.  Google Scholar

[19]

F. D. JiangN. S. Trudinger and X. P. Yang, On the Dirichlet problem for a class of augmented Hessian equations, J. Differential Equations, 258 (2015), 1548-1576.  doi: 10.1016/j.jde.2014.11.005.  Google Scholar

[20]

Q. N. JinY. Y. Li and H. Y. Xu, Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), 441-449.  doi: 10.4310/MAA.2005.v12.n4.a5.  Google Scholar

[21]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[22]

Y. Y. Li, Some existence results for fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[23]

J. K. LiuN. S. Trudinger and X. J. Wang, Interior $C^ {2, \alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.  doi: 10.1080/03605300903236609.  Google Scholar

[24]

G. Z. Lu and J. Y. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differential Equations, 258 (2015), 2054-2079.  doi: 10.1016/j.jde.2014.11.022.  Google Scholar

[25]

X. N. MaN. S. Trudinger and X. J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.  doi: 10.1007/s00205-005-0362-9.  Google Scholar

[26]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.   Google Scholar

[27]

N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.  Google Scholar

[28]

X. J. Wang, The $k$-Hessian equation, Geometric analysis and PDEs, 177–252, Lecture Notes in Math., 1977, Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-01674-5_5.  Google Scholar

show all references

References:
[1]

J. G. BaoX. H. Ji and H. G. Li, Existence and nonexistence theorem for entire subsolutions of $k$-Yamabe type equations, J. Differential Equations, 253 (2012), 2140-2160.  doi: 10.1016/j.jde.2012.06.018.  Google Scholar

[2]

L. A. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, Ⅰ. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar

[3]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅲ. Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

[4]

I. Capuzzo DolcettaF. Leoni and A. Vitolo, Entire subsolutions of fully nonlinear degenerate elliptic equations, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 147-161.   Google Scholar

[5]

I. Capuzzo DolcettaF. Leoni and A. Vitolo, On the inequality $F(x, D^2u)\geq f(u)+g(u)|Du|^q$, Math. Ann., 365 (2016), 423-448.  doi: 10.1007/s00208-015-1280-2.  Google Scholar

[6]

H. Car and R. Pröpper, Removable singularities of $m$-Hessian equations, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 6, 18 pp. doi: 10.1007/s00030-016-0429-3.  Google Scholar

[7]

K. S. Chou and X. J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016.  Google Scholar

[8]

D. P. Covei, The Keller-Osserman problem for the $k$-Hessian operator, arXiv: 1508.04653. Google Scholar

[9]

M. G. CrandallH. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[10]

A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 219-245.  doi: 10.1016/S0294-1449(00)00109-8.  Google Scholar

[11]

S. DumontL. DupaigneO. Goubet and V. Radulescu, Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.  doi: 10.1515/ans-2007-0205.  Google Scholar

[12]

P. L. Felmer and A. Quaas, On critical exponents for the Pucci's extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 843-865.  doi: 10.1016/S0294-1449(03)00011-8.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

B. Guan, Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.  Google Scholar

[15]

Y. HuangF. D. Jiang and J. K. Liu, Boundary $C^ {2, \alpha}$ estimates for Monge-Ampère type equations, Adv. Math., 281 (2015), 706-733.  doi: 10.1016/j.aim.2014.12.043.  Google Scholar

[16]

X. H. Ji and J. G. Bao, Necessary and sufficient conditions on solvability for Hessian inequalities, Proc. Amer. Math. Soc., 138 (2010), 175-188.  doi: 10.1090/S0002-9939-09-10032-1.  Google Scholar

[17]

F. D. Jiang and N. S. Trudinger, On the Dirichlet problem for general augmented Hessian equations, arXiv: 1903.12410. Google Scholar

[18]

F. D. JiangN. S. Trudinger and X. P. Yang, On the Dirichlet problem for Monge-Ampère type equations, Calc. Var. Partial Differential Equations, 49 (2014), 1223-1236.  doi: 10.1007/s00526-013-0619-3.  Google Scholar

[19]

F. D. JiangN. S. Trudinger and X. P. Yang, On the Dirichlet problem for a class of augmented Hessian equations, J. Differential Equations, 258 (2015), 1548-1576.  doi: 10.1016/j.jde.2014.11.005.  Google Scholar

[20]

Q. N. JinY. Y. Li and H. Y. Xu, Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), 441-449.  doi: 10.4310/MAA.2005.v12.n4.a5.  Google Scholar

[21]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[22]

Y. Y. Li, Some existence results for fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[23]

J. K. LiuN. S. Trudinger and X. J. Wang, Interior $C^ {2, \alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35 (2010), 165-184.  doi: 10.1080/03605300903236609.  Google Scholar

[24]

G. Z. Lu and J. Y. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differential Equations, 258 (2015), 2054-2079.  doi: 10.1016/j.jde.2014.11.022.  Google Scholar

[25]

X. N. MaN. S. Trudinger and X. J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.  doi: 10.1007/s00205-005-0362-9.  Google Scholar

[26]

R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647.   Google Scholar

[27]

N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.  Google Scholar

[28]

X. J. Wang, The $k$-Hessian equation, Geometric analysis and PDEs, 177–252, Lecture Notes in Math., 1977, Springer, Dordrecht, 2009. doi: 10.1007/978-3-642-01674-5_5.  Google Scholar

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